From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules

Fournier, Nicolas; Guillin, Arnaud

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 1, pp. 157-199.

Nous prouvons des résultats quantitatifs de convergence d'un système de particules conservatif vers la solution de l'équation de Landau homogène pour des potentiels durs. Il y a deux principales difficultés : (i) le caractère conservatif du système est un obstacle pour obtenir de l'indépendance (même approchée), comme c'est le cas pour de vrais systèmes de particules physiques ; (ii) les résultats connus de stabilité pour ces équations de Landau concernent des solutions régulières et paraissent difficiles à étendre pour étudier la vitesse de convergence de mesures empiriques. Pour le point (i), nous procédons à un double couplage. Nous couplons d'abord notre système avec des processus non linéaires non indépendants dont la loi résout en un certain sens l'équation de Landau. Nous construisons ensuite un second couplage afin de montrer que ces processus non linéaires ne sont pas loin d'être indépendants. Pour résoudre (ii), nous établissons de nouveaux résultats de stabilité pour l'équation de Landau pour des potentiels durs et des solutions de type mesure très générales. Finalement, en utilisant des idées de Rousset [26], nous montrons que dans le cas des molécules maxwelliennes, la convergence du système de particules est uniforme en temps.

We prove a quantitative result of convergence of a conservative stochastic particle system to the solution of the homogeneous Landau equation for hard potentials. There are two main difficulties: (i) the conservativeness of the particle system is an obstacle for approximate independence, as is the case for true physical particle systems; (ii) the known stability results for this class of Landau equations concern regular solutions and seem difficult to extend to study the rate of convergence of some empirical measures. Due to (i), we have to use a double-coupling. We first couple our particle system with some non independent nonlinear processes, of which the law solves, in some sense, the Landau equation. We then introduce a second coupling to prove that these nonlinear processes are not so far from being independent. To overcome (ii), we prove a new stability result for the Landau equation for hard potentials concerning very general measure solutions. Using finally some ideas of Rousset [26], we show that in the case of Maxwell molecules, the convergence of the particle system is uniform in time.

Classification : 82C40, 60K35, 65C05
Mots-clés : Landau equation, Uniqueness, Stability, Kac's particle system, Stochastic particle systems, Propagation of Chaos.

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